Real coverage probabilities differing significantly from advertised levels is a serious concern [1]}, [2]}.
A gap between real and advertised coverage probabilities can can have various causes, but here we focus on model misspecification. To avoid this misspecification bias, there are a few options: first, to take an approach that does not depend explicitly on a statistical model; second, to work with a model that is sufficiently broad that misspecification is virtually impossible; and third, to make some adjustments to correct for potential model misspecification. From a Bayesian perspective, the first fix is not available, since Bayes's formula requires a likelihood function. The second fix amounts to the use of Bayesian nonparametrics but, when the quantity of interest is a low-dimensional feature of the full distribution, introducing an infinite-dimensional parameter, with the computational and statistical challenges that entails, would be overkill. This leaves only the third option, but what kind of adjustments might the Bayesian consider? Recently, [3]} argued that a certain adjustment to the usual Bayesian posterior distribution could repair inconsistencies resulting from model misspecification. The goal of the present paper is to investigate the extent to which Grünwald and van Ommen's proposed adjustment—and other related adjustments in the literature—can close the gap between the real and advertised coverage probabilities of Bayesian posterior credible regions affected by model misspecification.
